Modification of kinetic theory for frictional spheres part II

Transcription

1 Modification of inetic theory for frictional phere part II Yang, L.; Padding, J.T.; Kuiper, J.A.M. Publihed in: Chemical Engineering Science DOI: 1.116/j.ce Publihed: /1/16 Document Verion Accepted manucript including change made at the peer-review tage Pleae chec the document verion of thi publication: A ubmitted manucript i the author' verion of the article upon ubmiion and before peer-review. There can be important difference between the ubmitted verion and the official publihed verion of record. People intereted in the reearch are advied to contact the author for the final verion of the publication, or viit the DOI to the publiher' webite. The final author verion and the galley proof are verion of the publication after peer review. The final publihed verion feature the final layout of the paper including the volume, iue and page number. Lin to publication Citation for publihed verion (APA): Yang, L., Padding, J. T., & Kuiper, J. A. M. (16). Modification of inetic theory for frictional phere part II: Model validation. Chemical Engineering Science, 15, DOI: 1.116/j.ce General right Copyright and moral right for the publication made acceible in the public portal are retained by the author and/or other copyright owner and it i a condition of acceing publication that uer recognie and abide by the legal requirement aociated with thee right. Uer may download and print one copy of any publication from the public portal for the purpoe of private tudy or reearch. You may not further ditribute the material or ue it for any profit-maing activity or commercial gain You may freely ditribute the URL identifying the publication in the public portal? Tae down policy If you believe that thi document breache copyright pleae contact u providing detail, and we will remove acce to the wor immediately and invetigate your claim. Download date: 8. May. 18

2 1 Modification of Kinetic Theory of Granular Flow for Frictional Sphere, Part II: Model validation L. (Lei) Yang, J.T. (Johan) Padding *, J.A.M. (Han) Kuiper Department of Chemical Engineering and Chemitry, Eindhoven Univerity of Technology, 56 MB, Eindhoven, The Netherland Correponding author : J.T.Padding@tue.nl Keyword: Fluidized granular bed; Rough particle; Multiphae flow; Two-fluid model; Dicrete particle model; PIV-DIA ABSTRACT The hydrodynamic of a dene olid-ga fluidized bed i tudied, uing a two fluid model (TFM) baed on our newly developed inetic theory of granular flow (KTGF) for rotating rough particle. The TFM imulation are validated by comparing with PIV-DIA experimental data (Buit et al., 14) and reult obtained from dicrete particle model (DPM) imulation of the bubbling fluidized bed. The TFM model prediction agree well with the experimental reult for the time-averaged particle axial velocity and olid volume fraction. The predicted level of the tranlational granular temperature and olid circulation pattern compare reaonably well with the reult obtained from the DPM imulation. The predicted rotational granular temperature in our TFM imulation how an almot uniform ditribution in the bed a a reult of the aumption that both the local mean rotational velocity and the gradient of the rotational granular temperature at the wall are zero, indicating direction for future improvement. A comparion between TFM imulation uing the preent KTGF model, and a more imple inetic theory for rapid flow of lightly frictional, nearly elatic phere derived by Jenin and Zhang (), i carried out to invetigate the influence of particle friction in the fluidized bed. The preent KTGF model lead to better agreement with DPM imulation and experimental reult for the axial particle velocity profile and olid volume fraction ditribution. 1. Introduction Ga-olid fluidized bed are widely employed in procee involving reaction, combution, eparation and claification in the petrochemical, pharmaceutical and energy indutrie. The deign and cale-up of thee indutrial device require a better undertanding of the dynamic of dene ga-olid flow. Owing to the enormou increae in computer power and algorithmic development, fundamental modelling of multiphae reactor ha become an effective tool to provide both qualitative and quantitative inight into complex ga-olid flow (Li et al. 5; van der Hoef et al., 6). To decribe the hydrodynamic of gaolid flow, Eulerian-Lagrangian model (Tuji et al., 1993; Hooman et al., 1996; Walton, 4) and Eulerian-Eulerian model (Gidapow, 1994; Enwald et al., 1996; Kuiper and van Swaaij, 1998) have been developed. Our dicrete particle model (DPM) i baed on the Eulerian-Lagrangian hard phere approach. In DPM the olid particle are treated explicitly and their motion i decribed by Newton econd law. The force on the particle arie from gravity, the interaction with the fluid phae, and colliion with other particle. For efficiency the averaged continuou fluid phae equation are olved on a grid which i much larger than the particle ize. A a reult, both a drag-force cloure and a colliion model have to be pecified for DPM. Detail of the DPM model and it application are given by Ritow (), Deen et al. (7), and Zhu et al. (7). The DPM can account for the particle-particle interaction in a fundamental and detailed manner. To undertand the capablity of the continuum TFM model, the DPM can be ued a

3 a numerical laboratory to validate the underlying aumption in cloure adopted for the olid phae in the KTGF theory. The two-fluid model (TFM) i baed on the Eulerian-Eulerian approach. In TFM the olid phae i treated a a econd continuum, interpenetrating with the continuou ga phae. Balance equation for ma, momentum and energy are olved by uing additional cloure equation for tre, vicoitie, thermal conductivitie, and energy diipation rate (Kuiper et al., 199; Wang et al., 1). The Euler- Euler approach ha emerged a a very promiing tool a a reult of it compromie between computational cot and amount of detail provided. The challenge of the model i to etablih an accurate hydrodynamic and rheological decription of the olid phae. State-of-the-art cloure have been obtained from the inetic theory of granular flow (Jenin & Savage, 1983; Ding & Gidapow, 199; Nieuwland et al., 1995). The earliet and mot widely ued KTGF model have been derived for dilute flow of inelatic, mooth, frictionle phere. However, real particle are uually rough and frictional, leading to particle rotation. More ophiticated model have been developed (Lun, 1991; Goldhtein and Shapiro, 1995; Jenin and Zhang, ; Kumaran, 6; Chialvo and Sundarean, 13; Zhao et al., 13) that conider particle roughne by including tangential retitution, coefficient of friction and rotational degree of freedom. However, until now there i no conenu on the bet general form for the inetic theory for rough phere accounting for the coefficient of friction, the coefficient of tangential retitution and particle rotation. In Part I, we have derived balance equation for three-dimenional rough phere, including cloure equation for the tranport coefficient. A ey finding wa that the particle tre ha an antiymmetric part which may be interpreted a a rotational vicoity. In thi wor, we implemented the new cloure equation in a three-dimenional extenion of the original two-dimenional model (in-houe code) developed by Kuiper et al. (199) and (1993). We will how that the reulting TFM imulation reult are in good agreement with PIV-DIA experimental data from Buit et al. (14) and imulation reult obtained with our in-houe DPM imulation. The imulated olid circulation pattern, averaged particle axial velocity and granular temperature profile are compared. Furthermore, a careful comparion i made between the preent model (Model A) and the effective model by Jenin and Zhang () (Model B) for the influence of particle friction.. Summary of inetic theory of rough phere The balance equation of the new KTGF model, including full expreion for the contitutive equation, are given in Table 1. In thi wor, the particle have rotational degree of freedom. During particle colliion not only linear momentum i tranferred but alo angular momentum. Detail of the model can be found in Part I. A more imple inetic theory for rapid flow of identical, lightly frictional, nearly elatic phere wa derived by Jenin and Zhang (). They introduced an effective coefficient of retitution e eff to account for the effect of particle friction on the energy diipation in cae the coefficient of friction i mall. The calculation of e eff can be found in Table. Table 1 Balance equation and contitutive equation for the frictional KTFG model. Balance equation: ( εgρg) + ( εgρgv g) = (T1) t ( εgρg) + ( εgρgv g) = (T) t

4 3 ( εgρgv g) + ( εgρgvv g g) = εg Pg ( εgτg) + εgρgg βa( vg v ) t (T3) ( ερ v) + ( ερ vv ) = ε Pg ( PI+ ετ) + ερ g+ βa( vg v ) t (T4) 3 ( ερ Θt) ( ερ t) :( P ε ) ε ( κt t) γt 3βA t + Θ = + + Θ Θ t (T5) 3 ( ερ Θr) ( ερ r) ε ( κr r) γr + Θ = Θ t (T6) Solid preure tenor: [ e εg ] P = ερθ 1+ (1 + ) (T7) t Bul vicoity: 4 Θt λ = ερσ g(1 + e) (T8) 3 π Solid tre tenor: T T τ = ( λ µ t )( v ) [ ( ) ] [ ( ) ] 3 I+ µ t v + v + µ r v v (T9) Ga tre tenor: T τg = ( λ g µ g)( vg) I+ µ g[ vg + ( v g) ] 3 (T1) Tranlational energy diipation rate: 19 Θ η (1 + η ) ( λ+ 1A ) + ( λ+ 1A ) γ + 1 v η (1 + η ) +5[ ( λ+ 1A ) ( λ+ 1A ) Rotational energy diipation rate: t t = Θtgρε σ π (T11) 96 t 5 5 γ r tgρε Θ A λa1 1 A4 λa3 σ π + =Θ v (T1) Tranlational hear vicoity: µ = µ 3 (1 + µ ) + t,.8 ( ),, 6 1 A1 tc tc 1 5 λ µ = ε g λ+ + η (T13-14) Rotational hear vicoity: Θt µ r = 8 ( λ+ 1) σgρε A 1 (T15) π Peudo tranlational thermal conductivity: 3 κt = κt ( 1 + κt ) + λ, c, t ( 1 16( 1) A1) c, κ = ε g η λ + (T16-17) Peudo rotational thermal conductivity:

5 4 = +, = ( + ) κr κ r κr c, L3 κ r ρ L 1 Θ Θ κ 16λ λ 1 g εσ A t t rc, 1 Θr π 3σε g t 5 A 1 5 L1 = Θ 1 A1 A 11 ( λ 1)( λ ) A1 ( 3λ 4) A9 5 π λ 3 3λ 3 (T18-19) 1 8 η1( + λ) L3 = (1 + gε) λθt + 5(1+ η1)a3 5 3(1 + λ) Radial ditribution function at contact (Ma and Ahmadi, 1986): g 1+.5ε ε ε 3 = 1+ 4ε ε 1 ε max (T) Where, A 1, A, A 3, A 4, A 6, A 7, A 9, A 11, A 1, A 13 are integral of trigonometric function and can be found 5Θr in Appendix B of Part I, and λ = i the ratio of rotational to tranlational granular temperature. Θ Table Effective coefficient of retitution (Jenin and Zhang, ) eeff = e a1+ a b a a t b µ µ µ 1 = πµ 1 arctan µ + 1 µ π 1+ µ µ 5µ π µ µ µ µ µ π ( 1+ µ ) 4 = 1 arctan + µ µ 1 = µ 1+ µ b µ π µ b = µ 1 arctan µ + µ π 1+ µ Θ r b1 7 µ ( 1+ e) =, µ = Θ b 1+ β t

6 5 3. Numerical imulation 3.1 Initial condition, boundary condition, and parameter etting The Euler-Euler imulation baed on the new KTGF model and the Euler-Lagrange imulation are carried out for a bubbling rectangular peudo-d fluidized bed, with particle and domain propertie lited in Table 3. No-lip boundary condition for the ide wall (left, right, front and bac ide of the rectangular domain) are ued for the ga phae. At the bottom inlet, a uniform ga velocity i pecified, wherea at the top outlet, atmopheric preure (11,35 Pa) i precribed. For the olid phae, partial lip boundary condition are ued on all wall. A relation for the olid velocity gradient and an expreion for the peudo Fourier fluctuation energy flux at the wall have been given by Sinclair and Jacon (1989), which ha been ued by Lu et al. (4) and Verma et al. (13) v εµ = z z t /3 r ε,max ε,max 3 πρ v α Θ ε Θ v α + e Θ κ = Θ r ε,max ε,max 1 ε ε t z 3(1 w) t t πρ 3 /3 t For the rotational granular temperature at the wall, we currently aume adiabatic condition, i.e. r κ Θ r =. Thi rather crude aumption i made becaue a theoretical treatment of the rotational r granular heat flux for our model i more involved and the topic of our current reearch. A initial condition, the olid volume fraction i et to.597 throughout the bed, olid velocitie are et to zero, and the ga velocity i et in accordance with the inflow boundary condition. Table 3 Propertie of particle and etting for model validation. Parameter DPM TFM Particle gla (ρ=56 g/m 3 ) gla (ρ=56 g/m 3 ) Particle diameter, σ 3 mm 3 mm Initial bed height -.3 m Domain ize m m Particle number 57 - Grid number (x y z ) Normal pring tiffne n=1 N/m Coefficient of normal retitution, e Coefficient of tangential retitution, β Coefficient of friction, µ Particle-wall colliion ε.1.1

7 6 Coefficient of normal retitution, e w Coefficient of tangential retitution, β w Coefficient of friction, μ w Specularity coefficient, α Initial bed voidage Simulation time 5 5 Superficial ga velocity, U g.35 m/, 3.8 m/.35 m/, 3.8 m/ Drag relation Ergun (195), and Wen & Yu (1966) Flow olver time tep Ergun (195), and Wen & Yu (1966) 3. Momentum exchange coefficient A proper drag model for the decription of the momentum exchange coefficient i vital for an adequate decription of fluidized bed. Many tudie (e.g. Wen and Yu, 1966; Van der Hoef et al., 5; Ozel et al., 9) have been dedicated to obtaining accurate drag relation. The drag model that i ued mot frequently in numerical model i a combination of the Ergun equation (195), originally developed for paced bed, at low poroitie, ε µ g ερ g βa = v, (for.8) g v εg εg σ σ and the Wen & Yu equation at high poroitie: 3 g 1.65 A C ερ β = d vg v εg, (for εg >.8) 4 σ Here the ingle phere drag coefficient i given by: ( Re ), (for Re 1) C d = Re.44, (for Re>1) which depend on the particle Reynold number Re = ε ρσ v v / µ. g g g g 3.3 Meaurement of particle height, granular temperature, and energy budget The time evolution of the averaged particle height h ( t ) can be ued to characterize the bed expanion dynamic. We will compare h ( t ) predicted by our TFM imulation with the one predicted by (more detailed) DPM imulation. In the latter cae, the averaged particle height follow from a numerical average: N part 1 h = hi N part i where h i i the height of particle i and N part i the total number of particle in the imulation domain. In TFM imulation, to obtain an equivalent etimate of the averaged particle height, we hould tae a weighted average, i.e. we hould weight by the olid volume fraction ε, of each cell :

8 7 h = Ncell Ncell ε, ε h, Where h i the height of cell, and N cell i the number of cell in the imulation domain. The granular temperature characterize the amount of random motion of individual particle in a mall region during a mall period. It i not only an important characteritic for the dynamic of ga-olid fluidization (Van der Hoef, 6), but alo an eential variable in TFM imulation becaue it determine the olid preure and tranport propertie. It i therefore important to compare the granular temperature predicted by our TFM model with the granular temperature meaured in DPM imulation. To thi end, in each computational cell of the DPM imulation, we firt determine the averaged tranlational and rotational olid velocitie of all N part, particle located in that cell: Npart, Npart, v 1 1 = v, ω = ω i i Npart, i Npart, i The tranlational granular temperature in cell i then computed a: N part, 1 Θ = ( v v ) t, i 3N part, i while the rotational granular temperature in cell i computed a: N part, I Θ = ( ω ω ) r, i 3mN part, i Finally, we compare the particle energy budget for the TFM (a decribed in Goldchmidt et al., 4) and DPM (a decribed in Van der Hoef et al., 6). We ditinguih between tranlational inetic energy E in, rotational granular energy E rotgran, and gravitational potential energy E pot. The tranlational inetic energy i further ubdivided into a convective contribution E conv (baed on the cell-averaged velocitie) and tranlational granular energy E gran (baed on the fluctuating velocitie). Detailed expreion are given in Table 4. Table 4 Expreion for the particle energy budget. Dicrete particle model N part 1 E in mv i i i - Two fluid model E conv N cell 1 mn v part, N cell 1 ε ρ v V,, cell,

9 8 N cell N 3 cell E gran mn part, Θt, ε, ρθt, Vcell, 3 N cell N 3 cell E rogrant mn part, Θr, ε, ρθr, Vcell, 3 E pot N part N cell mghi ε, ρghvcell, i Here V cell, i the volume of the cell. All the energy term in both DPM and TFM are calculated every.5 and averaged over a time pan of Reult and dicuion In thi part, the preent TFM model (model A) i validated by comparing with PIV-DIA experimental data from Buit et al. (14) and detailed data obtained from DPM imulation, focuing on particleparticle and ga-particle interaction. At the ame time, we compare the preent model with a TFM model (Model B) which i baed on mooth phere KTGF, but uing an effective coefficient of retitution a derived by Jenin and Zhang () (Table ). Fig. 1. Comparion of imulated average particle height, U g=.35 m/. Fig. 1 how the average particle height in the bed. The imulated reult from model A and model B are very imilar to thoe obtained from DPM, both in term of the average bed height and the fluctuation in bed height. However, the dynamic of the bed expanion predicted from model B i more pronounced than that from model A. The deviation can be attributed to the underetimation of energy loe during particle-particle colliion in model B. Next we focu on the olid circulation pattern. Due to coalecence, the bubble increae rapidly in ize a they move upward in the bed. A a conequence, a zone of increaed bubble development, initially cloe to the bottom wall near the ga inlet, i diplaced toward the center of the bed with

10 9 increaing height above the ga inlet. Particle appear to flow upward in region of more intene bubble activity and downward in region of leer bubble activity, which reult in the formation of a pronounced global olid circulation pattern with two ymmetric vortice at the top half of the bed. Fig. how the time-averaged olid circulation pattern from three model and the experiment for two different uperficial ga velocitie. The imulation reult from model A, model B and DPM how cloe agreement with experimental finding at a uperficial ga velocity of.35 m/. It can be oberved quantitatively that the olid in both model A and B move more vigorouly through the fluidized bed than predicted by the DPM imulation. When the uperficial ga velocity i increaed to 3.8 m/, the top vortice become elongated and a pronounced global olid circulation appear. Thi correpond well to obervation by many other (Laverman et al., 1; Goldchmidt et al., 4, Dan et al., 9) that the lateral movement of the bubble i enhanced at higher fluidization velocitie caued by the increaed circulation of the emulion phae. The macrocopic circulation pattern for the DPM and the TFM model are imilar. (a) (b) (c) (d) (e) (f) (g) Fig.. Time-averaged particle velocity from experiment and imulation. Upper row: U g=.35 m/, (a) PIV-DIA (Buit et al., 14); (b) Model A; (c) Model B; (d) DPM imulation; Lower row: U g=3.8 m/, (e) Model A; (f) Model B; (g) DPM imulation. Fig. 3 and 4 how a comparion of time-averaged axial particle velocity component at different uperficial ga velocitie. The axial particle velocity i poitive in the center and negative near the wall,

11 1 which indicate that particle move upward in the center and flow down near the wall. At the lower ga velocity (.35 m/), the imulated particle axial velocity from both model A and model B overpredict the acending and decending velocity magnitude. At higher ga velocity (3.8 m/), the predicted particle velocitie in the center agree well with the experiment and the DPM imulation at the height of.1 m. At larger height (z=. m and z=.3 m), the imulated particle velocity from both model A and model B are lower in magnitude than oberved in the experiment. There are deviation for the velocitie near the wall for all three model, including DPM, epecially at lower uperficial ga velocity. Van Buijtenen et al. (11) alo found uch deviation in pout fluidized bed. Still, both the imulation and the experiment how the ame velocity profile trend. Although the prediction for the velocity profile are imilar for model A and B, it can be een that in mot cae model A ha a omewhat better agreement with the DPM imulation than model B. The reader hould bear in mind that further improvement of model A i foreeen when updated boundary condition on the particle tre and tranlational and rotational granular temperature are included. z=.1 m z=. m z=.3 m Fig. 3. Comparion of averaged axial particle velocity for PIV-DIA and imulation in vertical direction at three cro-ection of the bed (5-5 ), U g=.35 m/.

12 11 z=.1 m z=. m z=.3 m Fig. 4. Comparion of time-averaged particle velocity for PIV-DIA and imulation in vertical direction at three cro-ection of the bed (5-5 ), U g=3.8 m/. Time-averaged olid fraction ditribution from the three model and the PIV-DIA experiment (Buit et al., 14) baed on the algorithm of van Buijtenen et al. (11) and de Jong et al. (1) are hown in Fig 5. We note that the PIV-DIA technique lac accuracy in the meaurement of the olid fraction, epecially at high olid fraction, o the comparion with experimental reult hould be done with care. All profile are qualitatively imilar except that the zone of intermediate denity (olid fraction ) move too much toward the bottom and the ide wall for model B, while thi intermediate denity zone remain at the bottom for model A and the more detailed DPM imulation. The dene zone (olid fraction.54 or higher) in both model A and model B are a little lifted toward the top of the bed. Overall, the extent of the dene zone predicted by model A i in good agreement with the extent predicted by DPM imulation and the experiment. On the other hand, the extent predicted by model B i much maller. In ummary, Fig. 5 again how that model A can provide better agreement with the DPM imulation and the experiment than model B.

13 1 (a) (b) (c) (d) Fig. 5. Comparion of time-averaged olid volume fraction for PIV-DIA (Buit et al., 14) and imulation (5-5 ), (a), PIV-DIA (Buit et al., 14); (b), DPM imulation; (c), Model A; (d), Model B, U g=.35 m/.

14 13 A comparion of the ditribution of tranlational granular temperature predicted by the three model i hown in Fig. 6 at uperficial ga velocity of.35 m/. A very clear difference between the DPM imulation on the one hand and both TFM imulation on the other hand can be oberved at height above.4 m, with the former predicting very low granular temperature and the latter very high granular temperature. Thi i an artifact caued by the fact that the region above.4 m i extremely dilute (ee Fig. 5). A a conequence in the DPM imulation often only or 1 particle are preent per computational cell, and hence no velocity fluctuation can be determined (which i then erroneouly recorded a zero granular temperature). On the other hand, becaue TFM i woring with continuou field, a granular temperature i imulated even in thee very dilute region. In our comparion between DPM and TFM we hould therefore exclude thi region above.4m. Having aid that, in TFM model A and B the zone with high tranlational granular temperature are till ituated mainly at the top of the bed, away from the ide wall. Model A predict a larger zone of high tranlational granular temperature than model B, however thi i influenced by the fact that particle rotation i not yet included in the boundary condition for rough ide wall in Model A, and therefore le energy i diipated during particle-wall colliion. From the freely bubbling experiment for Geldart D type particle of Boer (5), high value of tranlational granular temperature are found in the vicinity of the bubble. Goldchmidt et al. (4) pointed out that numerical imulation of peudo D fluidized bed predicted a lower granular temperature in area where the particle volume fraction wa high. Our imulation reult confirm thi finding. The tranlational granular temperature in all cae ha a imilar ditribution (below.4 m) and the overall magnitude of the tranlational granular temperature matche well. (a) (b) (c) Fig. 6. Contour plot of time-averaged (5-5 ) tranlational granular temperature (m / ) for TFM and DPM at U g=.35 m/, (a), DPM imulation; (b), Model A; (c), Model B.

15 14 Fig. 7 how a quantitative comparion of the tranlational granular temperature ditribution a a function of local olid fraction for the different model. The predicted tranlational granular temperature from all the model decreae with increaing of olid fraction. Thi decreae i mainly due to the increae in the colliion frequency between particle, leading to a tronger decreae in relative fluctuating velocitie. In the intermediate denity regime (olid fraction.5-.4), the imulated granular temperature from model A i higher than that of model B. In the dene regime (olid fraction >.45), where the majority of the particle reide, the oppoite i true: model A predict a lower granular temperature than model B. In thi dene limit model A i cloer to the DPM reult than model B, but both KTGF model over-predict the granular temperature. Again we emphaize that further improvement can be expected for model A with better boundary condition for the particle tre and tranlational and rotational granular temperature, which are epecially important for peudo-d ytem uch a tudied here. Fig. 7. Time-averaged (5-5 ) tranlational granular temperature a a function of olid fraction in the bed (ampled at height between.1 m and.35 m) fluidized at U g=.35 m/ for DPM (red), model A (blue), and model B (blac).

16 15 Fig. 8 how contour plot of the rotational granular temperature of TFM model A and DPM at uperficial ga velocity of.35 m/. The rotational granular temperature in TFM model A how an almot uniform ditribution in the bed, but the overall magnitude agree to a reaonable extent. In thi repect it i important to note that TFM model B ha a zero rotational granular temperature, which i further beide the truth. The reaon for the uniformity of the rotational granular temperature profile can be found in two major aumption made in the current verion of KTGF. Firtly, the mean rotational velocity i aumed to be zero, which mean that in the modelling of particle rotation only the rotational energy balance equation i olved. Secondly, the gradient of the rotational granular temperature at the wall i aumed to be zero (correponding to adiabatic wall for rotational granular temperature). It i clear that thee aumption may be too coare, epecially near boundarie which may in reality generate an average rotation velocity and generate or aborb rotational fluctuation. We are till woring on a careful theory for the boundary condition for the rotational granular temperature, but we find that preliminary imulation with uch improved boundary condition lead to a much more heterogeneou ditribution of the granular temperature throughout the fluidized bed, improving the agreement with DPM imulation. We will how thee reult in our next paper. (a) (b) Fig. 8. Contour plot of time-averaged (5-5 ) rotational granular temperature (m / ) for TFM and DPM at U g=.35 m/, (a): model A imulation, (b): DPM imulation.

17 16 Fig. 9 how the ratio of rotational to tranlational granular temperature a a function of the olid fraction for DPM and our model A at different uperficial ga velocitie. In the dilute region (olid volume fraction ), the predicted temperature ratio from model A i lower than that from DPM imulation. For higher olid volume fraction the temperature ratio imulated by model A i imilar to the DPM imulation. According to the inetic theory of Lun (1991), the ratio of rotational temperature to tranlational temperature i a function of the tangential coefficient of retitution only and independent of olid concentration. However, in the imulation of Lun and Bent (1994), the meaured temperature ratio in imple hear flow of teel phere with e =.93, β =, µ =.13 how a relatively pronounced dependence on olid concentration. Moreover, Jenin and Zhang () preented their analytic expreion of the temperature ratio for teady and unteady flow. In their theory, the temperature ratio i a function of normal and tangential retitution coefficient, and friction coefficient. Thi ratio i independent of the olid concentration, and alway underetimated compared to the imulation data of Lun and Bent (1994). Fig. 9. The time averaged (5-5 ) ratio of rotational to tranlational granular temperature λ predicted by model A (circle ) and DPM (quare ) at U g=.35 m/ and U g=3.8 m/ ampled in the bed at height between. m and.4 m. From Table 5 we find that the ditribution of the total energy of the particle acro the different contribution are almot the ame for TFM model A and DPM (for mall inlet ga velocity the deviation between the two model i 1.3%; for large inlet ga velocity the deviation i.6%). The contribution from the potential energy E pot and tranlational granular energy E gran become larger with increaing uperficial ga velocity, which i alo found in the imulation of Goldchmidt (4). For thi pecific ytem, it can be concluded that mot energy i preent a potential energy. However, TFM model B overpredict the total particle energy due to a too high potential particle energy, which i conitent with the increaed average particle height in Fig. 1. At the ame time, the amount of convective inetic energy i underetimated in TFM model B to a large extent. Both the high particle height and low convective inetic energy in model B are mainly caued by the abence of particle rotation. Therefore, we can conclude that the preent model A i uperior to model B in the prediction of the particle energy ditribution.

18 17 Table 5 Time averaged reult of energy balance for particle (1-5 ) Cae E tot [J] E in [J] E pot [J] E conv [J] E gran [J] E rotconv [J] E rotgran [J] DPM,.35 m/ (3.3%) 3.95(96.7%).18(3.1%).93(.%).8(.%).16(.4%) DPM, 3.8 m/ (7.94%) 4.9(9%).395(7.4%).36(.44%).16(.3%).4(.7%) Model A,.35 m/ (3.4%) 4.8(98.5%).19(.7%).18(.54%) -.33(.8%) Model B,.35 m/ (.%) 4.17(98%).7(1.4%).6(.6%) - - Model A, 3.8 m/ (5.7%) 5.(94.%).51(4.7%).5(1%) -.73(.13%) Model B, 3.8 m/ 5.4.1(3.9%) 5.1(96.1%).159(.9%).53(1%) - - Fig. 1 how the ditribution of the total energy diipation, a meaured in model A and model B, a a function of the olid concentration. The imulated total energy diipation from both model increae with increaing olid concentration from -.3, decreae when the olid concentration i in the range of.3-.4, and finally reache a plateau. The predicted total energy diipation from model A i larger than that from model B. From the expreion for the tranlational and rotational energy diipation (Eq. (T1) and (T11)), it can be een that both expreion have two term. The firt term relate to the colliional rate of inetic energy interchange. It incorporate the energy diipation from inelaticity and particle urface friction. The econd term include the velocity divergence. Thi term repreent energy loe due to compreion of granular material. The total energy diipation i the combination of thee two term. Model A conider not only the particle urface friction but alo particle rotation. A a reult, model A predict a larger amount of energy diipation. Fig. 1. Time-averaged (5-5 ) energy diipation over olid concentration with different model at U g=.35 m/ in the bed.

19 18 Finally, Fig. 11 how intantaneou naphot of bubble pattern at different time obtained by different model. Thee figure how that mall bubble are generated at the bottom, then bubble grow in ize due to coalecence, and move toward the center due to a lower reitance at the center (Kunii & Levenpiel, 1991). Comparing TFM model A and B, it can be oberved that both larger bubble and larger denely paced region are formed in TFM model A a a reult of the increaed energy diipation aociated with particle rotation. Although the particle ditribute more homogeneouly in the DPM imulation, lugging fluidization i oberved in both the DPM imulation and in the TFM model A imulation, while the phenomena in TFM model B are not clear. In concluion, TFM model A agree better with the DPM imulation than TFM model B, howing the importance of modelling the particle rotation

20 19 Fig. 11. Intantaneou naphot of ga fraction from different imulation at U g=.35 m/, top: DPM imulation, middle: TFM model A imulation, bottom: TFM model B imulation. 5. Concluion The hydrodynamic of a dene olid-ga fluidized bed ha been tudied uing inetic theory of granular flow (KTGF) for rough phere. The two-fluid model (TFM) imulation are validated by comparing with PIV-DIA experiment and DPM imulation of the ame dene olid-ga fluidized bed. We find that the TFM model with particle rotation (model A) agree well with PIV-DIA experimental reult of averaged particle axial velocity and olid volume fraction. Moreover, the predicted ditribution of the tranlational granular temperature in TFM model A agree cloely with the reult from DPM. Alo the olid circulation pattern compare reaonably well with the reult found in the DPM imulation. However, the predicted rotational granular temperature in the preent model A how an almot uniform ditribution in the bed a a reult of the aumption of zero mean rotational velocity and adiabatic boundary condition for the rotational granular temperature. Thu, although the preent model A can probably already be ued for the tudy of bul condition in dene olid-ga fluidized bed, proper boundary condition hould be derived and implemented for the rotational granular temperature and the rotational velocity, and the aumption of zero mean rotational velocity hould be relaxed. Thi will be the next topic of our reearch. We have performed comparion between the validated preent model A and the reult of a impler inetic theory for rapid flow of lightly frictional, nearly elatic phere derived by Jenin and Zhang () (model B). The imulation reult how that model A produce better agreement with DPM imulation reult in term of time-averaged axial particle velocity, olid volume fraction and the particle energy ditribution compared to model B. It can be concluded that the preent model improve the prediction obtained from the model derived by Jenin and Zhang (). Nomenclature I m n U mf moment of inertial ma of the particle, g particle number denity minimum fluidization velocity, m/

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